Discrete Dynamics in Nature and Society

Volume 2010 (2010), Article ID 408105, 19 pages

http://dx.doi.org/10.1155/2010/408105

## Robust Stability Analysis and Synthesis for Switched Discrete-Time Systems with Time Delay

School of Electrical and Control Engineering, Beijing University of Technology, Chaoyang District, Beijing 100124, China

Received 24 February 2010; Revised 24 June 2010; Accepted 18 July 2010

Academic Editor: Leonid Shaikhet

Copyright © 2010 Liguo Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The problems of robust stability analysis and synthesis for a class of uncertain switched time-delay systems with polytopic type uncertainties are addressed. Based on the constructive use of an appropriate switched Lyapunov function, sufficient linear matrix inequalities (LMIs) conditions are investigated to make such systems a uniform quadratic stability with an -gain smaller than a given constant level. System synthesis is to design switched feedback schemes, whether based on state, output measurements, or by using dynamic output feedback, to guarantee that the corresponding closed-loop system satisfies the LMIs conditions. Two numerical examples are provided that demonstrate the efficiency of this approach.

#### 1. Introduction

The stability analysis and synthesis problem of switched systems is one of the fundamental and challenging research topics, and various approaches has been obtained so far. For arbitrary switching law, a common Lyapunov function gives stability [1–3]. Liberzon and Hespanha have studied the global uniform asymptotic stability problem from the viewpoint of Lie algebra [4, 5]. On the other hand, Branicky [6], Johansen and Rantzer [7], respectively, have proposed the multiple Lyapunov function method for analysis and synthesis of switched systems with prescribed switching law. Furthermore, average dwell time technique [8–12] is an effective tool of choosing such switching law. In the recent literatures [13–16], stability analysis and synthesis of discrete time switched system were studied. A survey of switched systems problems have been proposed by Liberzon and Morse [5].

On the other hand, in many industrial hybrid systems such as power systems [17] and network control systems [13], time delay often occurs in the transition of the discrete states and the interior running of each subsystem. Therefore, more recently, much research attention has been devoted to the study of switched systems with time delay. In [18], -gain properties under arbitrary switching for a class of switched symmetric delay systems were studied. In [19], sufficient conditions of asymptotical stability were established for switched linear delay systems. Based on switched Lyapunov function approach, filtering problem of discrete-time switched systems with state delay was developed. In these papers mentioned above, only stable analysis is considered and state feedback results are proposed. In addition, output measurement and dynamic output feedback that are important synthesis methods for switched systems without delay are also expected to be effective for switched delay systems. However, no such results have been available up to now.

This paper studies the robust stability analysis and synthesis problems for uncertain switched delay systems with polytopic type uncertainties. Compared with the existing results on switched delay systems, the results of this paper have two features. Firstly, we give design of a novel switched Lyapunov function while the existing works commonly aim at the design of the common Lyapunov function. Secondly, dynamic output feedback is achieved while the existing literature usually addresses the state feedback control.

The rest of the paper is organized as follows. Section 2 briefly presents the uncertain switched time delay systems with polytopic type uncertainties and robust stability analysis with -gain, based on the switched Lyapunov function. Section 3 presents the robust synthesis with the switched state feedback or output feedback design schemes, while Section 4 reports the results for switched dynamic output feedback. Finally, the main conclusions are summarized in Section 5.

*Notation*

The notation used in this paper is fairly standard. The superscript stands for the matrix transposition, the notation refers to the Euclidea vector norm, denotes the dimensional Euclidean space. In addition, in the symmetric block matrices or long matrix expressions, we use * as an ellipsis for the terms that are introduced by symmetry, and stands for a block-diagonal matrix. A symmetric matrix means that is a positive (semipositive) definite matrix.

#### 2. Problem Formulation

Given a class of linear discrete time switched systems with time delay , , , , are the system state vector, control input, exogenous disturbance, measured output, and controlled output, respectively, and presents the time delay. The particular mode at any given time instant may be a selective procedure characterized by a switching rule of the following form: The function is usually defined using a partition of the continuous state space. Let denote the set of all selective rules. Therefore, the linear discrete time switched systems under consideration are composed of subsystems, each of which is activated at particular switching instant.

For a switching mode , the associated matrices contain uncertainties represented by a real convex-bounded polytopic model of the type where belongs to the unit simplex of vertices where are known as real constant matrices of appropriate dimensions.

Distinct from (2.1)–(2.3) is the free switched system

We have the following definitions.

*Definition 2.1. *Switched system (2.7)-(2.8) is Uniform Quadratic Stable (UQS), if there exist a Lyapunov functional , a constant , such that for all admissible uncertainties satisfying (2.5)-(2.6) and arbitrary switching rule activating subsystem at instant and subsystem at instant , the Lyapunov functional difference satisfies

*Definition 2.2. *Given a scalar , the -gain of switched system (2.7)-(2.8) over is

*Definition 2.3. *Switched system (2.7)-(2.8) is UQS with an -gain, if for all switching signal and for all admissible uncertainties satisfying (2.5)-(2.6), it is UQS and
We consider the following switching quadratic Lyapunov function:
where

Theorem 2.4. *The following statements are equivalent. *(1)*There exist a switching Lyapunov function of the type (2.12) with and a scalar such that switched system (2.7)-(2.8) with polytopic representation (2.5)-(2.6) is UQS with -gain.*(2)*There exist matrices , , , and a scalar satisfying the LMIs
**, *

*Proof. *(1)(2) Suppose that there exist a constant and a switching Lyapunov function of the type (2.12). Let the switching rule activates subsystem at instant and subsystem at instant . Thus,
Then, it can be shown that
where
Since , then .

By the Schur complement, is equivalent to
Applying the congruent transformation
we obtain
Upon using the vertex representation (2.5)-(2.6), we get (2.14) from (2.20).

(2)(1) Follow by reversing the steps in the proof and applying Definitions 2.1–2.3 to the system (2.7)-(2.8) for all modes and using (2.5)-(2.6).

The proof is complete.

Consider switched system (2.7)-(2.8) with , a special case of Theorem 2.4 is provided.

Corollary 2.5. *The following statements are equivalent.*(1)*There exists a switching Lyapunov function of type (2.12) with such that switched system (2.7)-(2.8) with , and polytopic representation (2.5)-(2.6) is UQS.*(2)*There exist matrices , and satisfying the LMIs
**, *

Since the exogenous disturbance in the switched system (2.7)-(2.8), we could let the coefficients , be zero matrices in the inequality (2.20). Directly deleting the third row and the third column from the matrix inequality (2.20), that is, deleting the term relevant to , we get the inequality (2.21).

#### 3. Switched Control Synthesis

Extending on Section 2, we examine here the problem of switched control synthesis using either switched state feedback or output feedback design schemes.

##### 3.1. Switched State Feedback

With reference to system (2.1)-(2.2), we consider that the arbitrary switching rule activates subsystem at an instant . Our objective herein is to design a switched state feedback such that the closed-loop system is UQS with an -gain .

Theorem 3.1. *Switched system (3.1)-(3.2) is UQS with an -gain, if there exist matrices , and a scalar , such that the following LMIs hold:
**
Moreover, the gain matrix is given by.*

*Proof. *It follows from Theorem 2.4 that switched system (3.1)-(3.2) is UQS with an -gain, if, for , there exist matrices , , and holding the following LMIs:

Applying the congruent transformation to LMIs (3.4), and let , we obtain
Upon using vertex representation (2.5)-(2.6), we get (3.3) from (3.5).

##### 3.2. Example

In this example, we design a switched state feedback control for this system based on Theorem 3.1, switching occurs between two modes described by the following coefficients.

Mode 1: consider

Mode 2: consider We set time delay , exogenous disturbance , then the feasible solution of LMIs (3.3) is given by , and Since , the control gains are With the initial state , and the following switching signal (Figure 1), the system state, control, and output signal are shown in Figures 2, 3, and 4, respectively.

From the simulation results, it can be clearly seen that the proposed control law guarantees the asymptotic stability of the closed-loop system.

##### 3.3. Switched Static Output Feedback

The objective of this subsection is to design switched output feedback , , such that the closed-loop system is UQS with an -gain .

Before developing our main results, we give the following lemma.

Lemma 3.2 (see [14]). *Given a matrix , and having the singular value decomposition form
**
where are unitary matrices and is a diagonal positive matrix. Let , then there exists a matrix , such that
**
if and only if
**
Thus, we get .*

Theorem 3.3. *Consider switched system (3.1)-(3.2) with polytopic representation (2.5)-(2.6) subject to the output feedback control and output matrix having the SVD form . The resulting closed-loop system is UQS with an -gain , if there exist matrices , , , , , , , and a scalar , such that the LMIs
**
have a feasible solution. Moreover, the feedback gain is given by , .*

*Proof. *In view of Theorem 2.4, we have
Let , then (3.15) can be rewritten as follows:
Upon using vertex representation (2.5)-(2.6), we get (3.14) from (3.16). According to Lemma 3.2, we obtain that .

##### 3.4. Example

Similar to Example , we wish to design a switched static output feedback control law based on Theorem 3.3. Using all coefficients in Example that adapt to Example .

Mode 1: consider

Mode 2: consider In this example, we set time delay , the exogenous disturbance , then the feasible solution of LMIs (3.14) is given by and Since , the control gains are With the initial state , and the switching signal as Example , the system state, control, and output signal are shown as follows.

#### 4. Switched Dynamic Output Feedback

We consider the more general case and employ at every mode , a switched dynamic output feedback scheme of the following form: Applying controller (4.1) to system (2.1)-(2.3) and letting , we get the closed-loop system where

Theorem 4.1. *Consider switched system (4.2) with polytopic representation (2.5)-(2.6) and output matrix having the SVD form , . This system is UQS with an -gain , if there exist
**
and a scalar satisfying the systems LMIs
**
Moreover, the gain matrix is given by and *

*Proof. *In view of Theorem 2.4, we have
By applying (4.3)-(4.4) to (4.6), we obtain
Let and using vertex representation (2.5)-(2.6), we get (4.5) from (4.7).

#### 5. Conclusions

We have examined -gain analysis and control synthesis for a class of switched time-delay systems with convex-bounded parameter uncertainties in the system matrices using an appropriate switched Lyapunov functional. LMI-based feasibility conditions have been developed to ensure that the linear switched time-delay system with polytopic uncertainties is UQS with an -gain. Switched feedback schemes have been designed using state output measurements and by using dynamic output feedback to guarantee that the corresponding closed-loop system enjoys the uniform quadratic stability with an -gain smaller than a prescribed constant level.

#### Acknowledgments

The authors are grateful to the comments of anonymous referees and remarks of the editors. This work was supported by the Grants from NSFC (nos. 60774037, 60904069, 10926048, and 60911120067), the Doctoral Fund of Ministry of Education of China (no. 20091103120008), and the Science and Technology Development Programs of Beijing Educational Commission (no. KM201010005031).

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